According to
Ashitaka of the Lain mailing list, the Kanji sound for 'Rein' means 'the sound of jewels. 'Iwakura' is the name of a
samurai group serving under Iwakura Tomomi. The Meji emperor sent the samurai to Europe to study European culture in 1872-73.
Gn0sis informs me that Iwakura literally means: 'Stone Warehouse'.
These are the names of Iwakura Lain in Japanese:
Romaji: Iwakura Rein
Kanji:岩倉玲音
Hiragana: ぃわくられぃん
Katakana:ィヮクラレィン
Thanks to Ashitaka of the Lain mailinglist.